Reducing Circuit Resources in Grover's Algorithm via Constraint-Aware Initialization

TL;DR

This paper introduces a systematic framework for constraint-aware initialization in Grover's algorithm, reducing circuit resources by encoding problem constraints, demonstrated through linear constraints and the exact-cover problem.

Contribution

It presents a simple preprocessing method for structured initial states in Grover's algorithm, improving resource efficiency in terms of gate counts and circuit depth.

Findings

Constraint-aware initialization reduces oracle queries needed.

Structured initial states lower circuit depth and gate counts.

Numerical validation on the exact-cover problem supports efficiency gains.

Abstract

Grover's search algorithm provides a quadratic speedup over classical brute-force search in terms of query complexity and is widely used as a versatile subroutine in numerous quantum algorithms, including those for combinatorial problems with large search spaces. For such problems, it is natural to reduce the effective search space by incorporating problem constraints at the initialization step, which in Grover's algorithm can be achieved by preparing structured initial states that encode constraint information. In this work, we present a systematic framework with a simple preprocessing procedure for constraint-aware initialization in Grover's algorithm, focusing on problems with linear constraints. While such structured initial states can reduce the number of oracle queries required to obtain a solution, their preparation incurs additional circuit-level costs. We therefore offer a conservative circuit-level resource analysis, showing that the resulting constraint-aware initialization can improve resource efficiency in terms of gate counts and circuit depth. The validity of the framework is further demonstrated numerically using the exact-cover problem. Overall, our results indicate that this approach serves as a practical baseline for achieving more resource-efficient implementations of Grover's algorithm compared to the standard uniform initialization.